A154637 a(n) is the ratio of the sum of squares of the bends of the circles that are added in the n-th generation of Apollonian packing, to the sum of squares of the bends of the initial three circles.

1, 2, 66, 1314, 26082, 517698, 10275714, 203961186, 4048396578, 80356048002, 1594975770306, 31658447262114, 628384017931362, 12472705016840898, 247568948283023874, 4913960850609954786, 97536510167350024098, 1935988320795170617602, 38427156885401362279746, 762735172745641733742114

Offset: 0

Author

Colin Mallows, Jan 13 2009

Keywords

• easy
• nonn

Comments

For more references and links, see A189226.

Examples

			Starting with three circles with bends -1,2,2, the ssq is 9. The first derived generation has two circles, each with bend 3. So a(1) = (9+9)/9 = 2.
		

Links

• Colin Barker, Table of n, a(n) for n = 0..750
• Colin Mallows, Growing Apollonian packings, J. Integer Sequences v.12, article 09.2.1 (2009).
• Index entries for linear recurrences with constant coefficients, signature (20,-3).

Crossrefs

For starting with four circles, see A137246. For sums of bends, see A135849 and A154636. For three dimensions, see A154638 - A154645.

Cf. also A189226, A189227.

Programs

Mathematica

CoefficientList[Series[(29 z^2 - 18 z + 1)/(3 z^2 - 20 z + 1), {z, 0, 100}], z] (* and *) LinearRecurrence[{20, -3}, {1, 2, 66}, 100] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)

PARI

Vec((1-18*x+29*x^2)/(1-20*x+3*x^2) + O(x^30)) \\ Colin Barker, Nov 16 2016

Formula

G.f.: (1-18*x+29*x^2) / (1-20*x+3*x^2).

From Colin Barker, Nov 16 2016: (Start)

a(n) = ((133-13*sqrt(97))*(10+sqrt(97))^n - (10-sqrt(97))^n*(133+13*sqrt(97))) / (3*sqrt(97)) for n>0.

a(n) = 20*a(n-1) - 3*a(n-2) for n>2.

(End)

Extensions

More terms from N. J. A. Sloane, Nov 22 2009